For example, if the mean value m of the reference population is 100, and if a sample provides 17 data points (df = 16), with a mean value of 90 and a standard deviation s of 10, we calculate for |t| a value of 4. If we adopt for α a value of 0.05, the table provides for tc a value of 2.12 (two-tailed test). We reject therefore the null hypothesis since |t| > tα, and conclude that the mean of the population from which the sample was drawn is significantly different (p < 0.05) from the reference mean m.
Two-sample t-test. The t-distribution may also be used to test whether the means of two populations from which samples are drawn are the same. The means of the populations are mx and my, respectively. The number of data points in each sample is Nx and Ny, and the corresponding sample means are and . The sample variances are and . The t-value is computed as
with degrees of freedom df = Nx + Ny – 2. We test again the null hypothesis (i.e., the hypothesis that mx = my) by comparing the t-value with the critical value tc. If |t| > tα the null hypothesis is discarded, and the difference in the means of the two populations is significant.
Critical values of Student’s t-distribution with different degrees of freedom (df)
One-tail
Two-tails 0.50
1.00 0.25
0.50 0.20
0.50 0.15
0.30 0.10
0.20 0.05
0.10 0.025
0.05 0.01
0.02 0.005
0.01 0.001
0.002 0.0005
0.001
df
1 0.000 1.000 1.376 1.963 3.078 6.314 12.71 31.82 63.66 318.31 636.62
2 0.000 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 22.327 31.599
3 0.000 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 10.215 12.924
4 0.000 0.741 0.941 1.190 1.533 2.132 1.776 3.747 4.604 7.173 8.610
5 0.000 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 5.893 6.869
6 0.000 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 0.000 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 0.000 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 0.000 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.297 4.781
10 0.000 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 0.000 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 0.000 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 0.000 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 0.000 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 0.000 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 0.000 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 0.000 0.689 0.863 1.069 1.33 1.740 2.110 2.567 2.898 3.646 3.965
18 0.000 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 0.000 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 0.000 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.552 3.850
21 0.000 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.527 3.819
22 0.000 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.505 3.792
23 0.000 0.685 1.060 1.319 1.714 2.069 2.500 2.807 3.485 3.485 3.768
24 0.000 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.467 3.745
25 0.000 0.684 0.856 1.056 1.316 1.708 2.060 2.485 2.787 3.450 3.725
26 0.000 0.684 0.856 1.056 1.315 1.706 2.056 2.479 2.779 3.435 3.707
27 0.000 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.421 3.690
28 0.000 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.408 3.674
29 0.000 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.396 3.659
30 0.000 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.385 3.646
40 0.000 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.307 3.551
60 0.000 0.679 0.848 1.045 1.296 1.671 2.000 2.390 2.660 3.232 3.460
80 0.000 0.678 0.846 1.043 1.292 1.664 1.990 2.374 2.639 3.195 3.416
100 0.000 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.174 3.390
1000 0.000 0.675 0.842 1.037 1.282 1.646 1.962 2.330 2.581 3.098 3.300
∞ 0.000 0.674 0.842 1.036 1.282 1.645 1.960 2.326 2.576 3.090 3.290
Confidence level 0% 50% 60% 70% 80% 90% 95% 98% 99% 99.8% 99.9%
References
Co T. B. (2013) Methods of Applied Mathematics for Engineers and Scientists, Cambridge University Press, Cambridge.
Durran D. R. (2010) Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin.
Press W. H., Teukolsky S. A., Vetterling W. T., and Flannery B. P. (2007) Numerical Recipes: The Art of Scientific computing, Cambridge University Press, Cambridge.
Further Reading
Earth System and Climate Science
Baird C. and Cann M. (2008) Environmental Chemistry, Bookman, Taipei.
Houghton J. (2009) Global Warming: The Complete Briefing, Cambridge University Press, Cambridge.
Kump L. R., Kasting J. F., and Crane R. G. (2010) The Earth System, Prentice Hall, Upper Saddle River, NJ.
Marshall J. and Plumb R. A. (2008) Atmosphere, Ocean and Climate Dynamics: An Introductory Text, Academic Press, New York.
Atmospheric Science
Frederick J. E. (2008) Principles of Atmospheric Science, Jones & Bartlett Learning, Sudbury, MA.
Goody R. (1995) Principles of Atmospheric Physics and Chemistry, Oxford University Press, Oxford.
Salby M. L. (2012) Physics of the Atmosphere and Climate, Cambridge University Press, Cambridge.
Wallace J. M. and Hobbs P. V. (2006) Atmospheric Science: An Introductory Survey, Academic Press, New York.
Atmospheric Chemistry
Barker, J. R. (ed.) (1995) Progress and Problems in Atmospheric Chemistry, World Scientific, River Edge, NJ.
Brasseur, G. P., Orlando J. J., and Tyndall G. S. (eds.) (1999) Atmospheric Chemistry and Global Change, Oxford University Press, Oxford.
Brasseur G. P., Prinn R. G., and Pszenny A. A. P. (eds.) (2003) Atmospheric Chemistry in a Changing World: An Integration and Synthesis of a Decade of Tropospheric Chemistry Research, Springer, New York.
Burrows, J. P., Platt U., and Borrell P. (eds.) (2011) The Remote Sensing of Tropospheric Composition from Space, Springer, New York.
Finlayson-Pitts B. J, and Pitts Jr. J. N. (2000) Chemistry of the Upper and Lower Atmosphere: Theory, Experiments, and Applications, Academic Press, New York.
Hobbs P. V. (1995) Basic Physical Chemistry for the Atmospheric Sciences, Cambridge University Press, Cambridge.
Hobbs P. V. (2000) Introduction to Atmospheric Chemistry, Cambridge University Press, Cambridge.
Jacob D. J. (1999) Introduction to Atmospheric Chemistry, Princeton University Press, Princeton, NJ.
Jaeschke, W. (ed.) (1986) Chemistry of Multiphase Atmospheric Systems, Springer-Verlag, Berlin.
Seinfeld J. H. and Pandis S. N. (2006) Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Wiley, New York.
Singh, H. B. (ed.) (1995) Composition, Chemistry, and Climate of the Atmosphere, Van Nostrand Reinhold, New York.
Sportisse B. (2008) Fundamentals in Air Pollution. From Processes to Modelling, Springer, New York.
Warneck P. (1999) Chemistry of the Natural Atmosphere, Academic Press, New York.
Warneck P. and Williams J. (2012) The Atmospheric Chemist’s Companion: Numerical Data for Use in the Atmospheric Sciences, Springer, New York.
Wayne R. P. (1985) Chemistry of Atmospheres: An Introduction to the Chemistry of the Atmospheres of Earth, the Planets, and their Satellites, Oxford University Press, Oxford.
Yung Y. L. and DeMore W. B. (1999) Photochemistry of Planetary Atmospheres, Oxford University Press, Oxford.
Atmospheric Physics and Dynamics
Andrews D. G. (2010) An Introduction to Atmospheric Physics, Cambridge University
Press, Cambridge.
Holton J. R. (2004) An Introduction to Dynamic Meteorology, Academic Press, New York.
Houghton J. (2002) The Physics of Atmospheres, Cambridge University Press, Cambridge.
Mak M. (2011) Atmospheric Dynamics, Cambridge University Press, Cambridge.
McWilliams J. C. (2006) Fundamentals of Geophysical Fluid Dynamics, Cambridge University Press, Cambridge.
Neufeld Z. and Hernández-Garcia E. (2010) Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach, Imperial College Press, London.
Pruppacher H. R., and Klett J. D. (1997) Microphysics of Clouds and Precipitation, Kluwer, Dordrecht.
Riegel C. A. and Bridger A. F. C. (1992) Fundamentals of Atmospheric Dynamics and Thermodynamics, World Scientific, River Edge, NJ.
Salby M. L. (1996) Fundamentals of Atmospheric Physics, Academic Press, New York.
Vallis G. K. (2006) Atmospheric and Oceanic Fluid Dynamics, Fundamentals and Large-Scale Circulation, Cambridge University Press, Cambridge.
Radiative Transfer
Goody R. M. and Yung Y. L. (1989) Atmospheric Radiation, Theoretical Basis, Oxford University Press, Oxford.
Liou K. N. (2002) An Introduction to Atmospheric Radiation, Academic Press, New York.
Petty G. W. (2006) A First Course in Atmospheric Radiation, Sundog Publishing, Madison, WI.
Thomas G. E., and Stamnes K. (1999) Radiative Transfer in the Atmosphere and Ocean, Cambridge University Press, Cambridge.
Turbulence, Boundary Layer Meteorology, and Surface Exchanges
Bonan G. (2008) Ecological Climatology, Concepts and Applications, Cambridge University Press, Cambridge.
Garratt J. R. (1992) The Atmospheric Boundary Layer, Cambridge University Press, Cambridge.
Granier, C., Artaxo P., and Reeves C. E. (eds.) (2004) Emissions of Atmospheric Trace Compounds, Kluwer, Dordrecht.
Monson R. and Baldocchi D. (2009) Terrestrial Biosphere–Atmosphere Fluxes, Cambridge University Press, Cambridge.
Stull R. B. (1988) An Introduction to Boundary Layer Meteorology, Kluwer, Dordrecht.
Wyngaard J. C. (2010) Turbulence in the Atmosphere, Cambridge University Press, Cambridge.
Vilà-Guerau de Arellano J., van Heerwaarden C. C., van Stratum B. J. H., and van den Dries K. (2015) Atmospheric Boundary Layer: Integrating Air Chemistry and Land Interactions, Cambridge University Press, Cambridge.
Middle Atmosphere
Andrews D. G., Holton J. R., and Leovy C. B. (1987) Middle Atmosphere Dynamics, Academic Press, New York.
Brasseur G. P. and Solomon S. (2005) Aeronomy of the Middle Atmosphere: Chemistry and Physics of the Stratosphere and Mesosphere, Springer, New York.
Dessler A. (2000) Chemistry and Physics of Stratospheric Ozone, Academic Press, New York.
Müller, R. (ed.) (2012) Stratospheric Ozone Depletion and Climate Change, RSC Publishing, Cambridge.
Numerical Methods, Modeling, and Data Assimilation
Co T. B. (2013) Methods of Applied Mathematics for Engineers and Scientists, Cambridge University Press, Cambridge.
Courant R. and Hilbert D. (1962), Methods of mathematical Physics, vols. 1 and 2, Wiley, New York.
Daley R. (1991) Atmospheric Data Analysis, Cambridge University Press, Cambridge.
DeCaria A. J. and Van Knowe G. E. (2014) A First Course in Atmospheric Numerical Modeling, Sundog Publishing, Madison, WI.
Durran D. R. (2010) Numerical Methods for Fluid Dynamics, with Applications to Geophysics, Springer, New York.
Gear W. (1971) Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ.
Fox R. O. (2003) Computational Models for Turbulent Reacting Flows, Cambridge University Press, Cambridge.
Gershenfeld N. (1999) The Nature of Mathematical Modeling, Cambridge University Press, Cambridge.
Hairer E. and Wanner G. (1996) Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin.
Jacobson M. Z. (1999) Fundamentals of Atmospheric Modeling, Cambridge University Press, Cambridge.
Kalnay E. (2003) Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, Cambridge.
Kiehl, J. T. and Ramanathan V. (eds.) (2006) Frontiers of Climate Modeling, Cambridge University Press, Cambridge.
Lahoz, W., Khattatov B., and Ménard R. (eds.) (2010) Data Assimilation: Making Sense of Observations, Springer, New York.
Lauritzen, P. H., Jablonowski C., Taylor M. A., and Nair R. D. (eds.) (2011) Numerical Techniques for Global Atmospheric Models: Tutorials, Springer, New York.
Müller P. and von Storch H. (2004) Computer Modelling in Atmospheric and Oceanic Sciences: Building Knowledge, Springer, New York.
Potter D. (1977) Computational Physics, Wiley, New York.
Press W. H., Teukolsky S. A., Vetterling W. T., and Flannery B. P. (2007) Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge.
Rodgers C. D. (2000) Inverse Methods for Atmospheric Sounding: Theory and Practice, World Scientific, River Edge, NJ.
Slingerland R. and Kump L. (2011) Mathematical Modeling of Earth’s Dynamical Systems: A Primer, Princeton University Press, Princeton, NJ.
Stensrud D. J. (2007) Parameterization Schemes: Keys to Understanding Numerical Weather Prediction Models, Cambridge University Press, Cambridge.
Stoer J. and Bulirsch R. (1993) Introduction to Numerical Analysis, 2nd edition, Springer-Verlag, Berlin.
Trenberth K. E. (ed.) (1992) Climate System Modeling, Cambridge University Press, Cambridge.
Warner T. T. (2011) Numerical Weather and Climate Prediction, Cambridge University Press, Cambridge.
Washington W. M. and Parkinson C. L. (2005) An Introduction to Three-Dimensional Climate Modeling, University Science Books, Sausalito, CA.
Wilks D. S. (2011) Statistical Methods in the Atmospheric Sciences, Academic Press, New York.
Index
absorption cross-section, 207
absorption of radiation absorption efficiency, 219
definition, 207
absorptivity, 212
acceleration due to gravity. See standard gravity
acetone air–sea exchange, 432
acid rain, 14
actinic flux, 210actinic flux density, 210
actinic photon flux, 210
activated complex, 230
activation energy, 230
ADI. See alternating direction implicit method
adiabatic lapse rate and atmospheric stability, 35
dry, 33
wet, 34
adjoint, 502–509continuous, 509
discrete, 509
forcing, 502, 521
self-adjoint, 509
sensitivities, 504–505
advection, 275, 277equation, 276–277, 279, 281–282, 303, 311
semi-Lagrangian, 276
timescale, 280
advection–diffusion equation, 139
aeronomy, 26–27
aerosol microphysics, 95
aerosol observations atmospheric components measured, 440–441
in-situ composition, 444
size distribution, 443
total concentration, 442
remote aerosol optical depth (AOD), 460
aerosols accumulation mode, 244
and air quality, 14
Aitken nuclei mode, 244
atmospheric abundance, 21
chemical composition, 78–80types, 78
and climate, 14
cloud condensation nuclei (CCN), 81
and cloud formation, 29
coarse mode, 244
core-shell model, 80
hydrophobic, hydrophilic, 81
hygroscopicity, 80
microphysical processes, 243schematic representation, 245
mixing state, 80
nucleation mode, 244
optical properties aerosol optical depth, 82
>
scattering, absorption, extinction efficiency, 81
primary (POA) and secondary organic aerosol (SOA), 80
size distribution, 21, 76–78discrete representation, 247
modal representation, 249
modes, 77–78
monodisperse, polydisperse, 76
monodisperse representation, 247
sectional representation, 248
size distribution functions, 76–77, 243
spline representation, 247
sulfate–nitrate–ammonium (SNA) aerosol, 78, 79
terminology, 75
aggregation bias, 518
aggregation error, 516–520covariance matrix, 518
aggregation matrix, 517
air density, 21
air mass factor (AMF), 215, 459
air parcel definition, 32
air pressure. See atmospheric pressure
air temperature. See atmospheric temperature
aircraft measurements. See observing platforms
albedo of Earth’s surface, 223
aliasing, 140
alternating direction implicit method, 178
AMF. See air mass factor (AMF)
angular rotation velocity of Earth, 36
anti-cyclone, 38in the subtropics, 44
AO. See Arctic Oscillation (AO)
Ar. See Argon (Ar)
Arctic Oscillation (AO), 47and middle atmosphere dynamics, 51
area sources, 412
Modeling of Atmospheric Chemistry Page 71